Problem: All of the 3rd grade teachers and students from Gardner Bullis went on a field trip to an art museum. Tickets were $$7.50$ each for teachers and $$4.00$ each for students, and the group paid $$50.50$ in total. The next month, the same group visited a natural history museum where the tickets cost $$30.00$ each for teachers and $$7.50$ each for students, and the group paid $$142.50$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${7.5x+4y = 50.5}$ ${30x+7.5y = 142.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-30x-16y = -202}$ ${30x+7.5y = 142.5}$ Add the top and bottom equations together. $ -8.5y = -59.5 $ $ y = \dfrac{-59.5}{-8.5}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {7.5x+4y = 50.5}$ to find $x$ ${7.5x + 4}{(7)}{= 50.5}$ $7.5x+28 = 50.5$ $7.5x = 22.5$ $x = \dfrac{22.5}{7.5}$ ${x = 3}$ You can also plug ${y = 7}$ into $ {30x+7.5y = 142.5}$ and get the same answer for $x$ ${30x + 7.5}{(7)}{= 142.5}$ ${x = 3}$ There were $3$ teachers and $7$ students on the field trips.